f(x)=x^3
Question
f\left(x\right)=x^{3}
Function
Find the inverse
f^{-1}\left(x\right) = x^{\frac{1}{3}}
Evaluate
f\left(x\right)=x^{3}
\text{In the equation for }f\left(x\right)\text{,replace }f\left(x\right)\text{ with }y
y=x^{3}
\text{Interchange }x\text{ and }y
x=y^{3}
Swap the sides
y^{3}=x
\text{Raise both sides of the equation to the }\frac{1}{3}\text{-th power to eliminate the isolated }\frac{1}{3}\text{-th root}
\left(y^{3}\right)^{\frac{1}{3}}=x^{\frac{1}{3}}
Evaluate the power
y=x^{\frac{1}{3}}
Solution
f^{-1}\left(x\right) = x^{\frac{1}{3}}
Evaluate the derivative
f^{\prime}\left(x\right)=3x^{2}
Evaluate
f\left(x\right)=x^{3}
Take the derivative of both sides
f^{\prime}\left(x\right)=\frac{d}{dx}\left(x^{3}\right)
Solution
f^{\prime}\left(x\right)=3x^{2}
Find the domain
x \in \mathbb{R}
Evaluate
f\left(x\right)=x^{3}
Solution
x \in \mathbb{R}
\text{Find the }x\text{-intercept/zero}
x=0
Evaluate
f\left(x\right)=x^{3}
\text{To find the }x\text{-intercept,set }f\left(x\right)\text{=0}
0=x^{3}
Evaluate
x^{3}=0
Solution
x=0
Find the y-intercept
f\left(0\right)=0
Evaluate
f\left(x\right)=x^{3}
\text{To find the }y\text{-intercept,set }x\text{=0}
f\left(0\right)=0^{3}
Solution
f\left(0\right)=0
Find the critical numbers
x=0
Evaluate
f\left(x\right)=x^{3}
Find the domain of the function
f\left(x\right)=x^{3},x \in \mathbb{R}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
f^{\prime}\left(x\right)=3x^{2}
Find the domain of the derivative within the domain of the original function
f^{\prime}\left(x\right)=3x^{2},x \in \mathbb{R}
\text{Substitute }f^{\prime}\left(x\right)=0\text{ to find the critical numbers}
0=3x^{2}
Swap the sides
3x^{2}=0
Divide both sides
\frac{3x^{2}}{3}=\frac{0}{3}
Simplify
\frac{3x^{2}}{3}=0
Simplify
x^{2}=0
Solution
x=0
Find the local extrema
\textrm{No local extrema}
Evaluate
f\left(x\right)=x^{3}
Find the domain of the function
f\left(x\right)=x^{3},x \in \mathbb{R}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
f^{\prime}\left(x\right)=3x^{2}
Find the domain of the derivative within the domain of the original function
f^{\prime}\left(x\right)=3x^{2},x \in \mathbb{R}
\text{Substitute }f^{\prime}\left(x\right)=0\text{ to find the critical numbers}
0=3x^{2}
Swap the sides
3x^{2}=0
Divide both sides
\frac{3x^{2}}{3}=\frac{0}{3}
Simplify
\frac{3x^{2}}{3}=0
Simplify
x^{2}=0
Calculate
x=0
Determine the intervals around the critical point
\begin{align}&\left(-\infty,0\right),\left(0,+\infty\right)\end{align}
Choose one point from each interval
x_{1}=-1,x_{2}=1
Find the values of the derivatives for the chosen point
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Evaluate
f^{\prime}\left(-1\right)=3\left(-1\right)^{2}
Evaluate the power
f^{\prime}\left(-1\right)=3\times 1
Multiply the numbers
f^{\prime}\left(-1\right)=3
f^{\prime}\left(-1\right)=3,x_{2}=1
Find the values of the derivatives for the chosen point
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Evaluate
f^{\prime}\left(1\right)=3\times 1^{2}
Evaluate the power
f^{\prime}\left(1\right)=3\times 1
Multiply the numbers
f^{\prime}\left(1\right)=3
f^{\prime}\left(-1\right)=3,f^{\prime}\left(1\right)=3
\text{Since the derivative is positive for }x<0\text{ and positive for }x>0\text{,the function has no local extrema at }x=0
\begin{align}&\text{No local extrema at }x=0\end{align}
Solution
\textrm{No local extrema}
Find the increasing or decreasing interval
\begin{align}&\text{The increasing interval is }x \in \mathbb{R}\\&\textrm{No decreasing interval}\end{align}
Evaluate
f\left(x\right)=x^{3}
Find the domain of the function
f\left(x\right)=x^{3},x \in \mathbb{R}
\text{Use }\frac{d}{dx} x^{n}=n x^{n-1}\text{ to find derivative}
f^{\prime}\left(x\right)=3x^{2}
Find the domain of the derivative within the domain of the original function
f^{\prime}\left(x\right)=3x^{2},x \in \mathbb{R}
\text{Substitute }f^{\prime}\left(x\right)=0\text{ to find the critical numbers}
0=3x^{2}
Swap the sides
3x^{2}=0
Divide both sides
\frac{3x^{2}}{3}=\frac{0}{3}
Simplify
\frac{3x^{2}}{3}=0
Simplify
x^{2}=0
Calculate
x=0
Determine the intervals according to the critical numbers and the domain of original function
\begin{align}&x\leq 0\\&x\geq 0\end{align}
Choose one point from each interval
\begin{align}&x_{1}=-1\\&x_{2}=1\end{align}
Find the values of the derivatives for the chosen point
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Evaluate
f^{\prime}\left(-1\right)=3\left(-1\right)^{2}
Evaluate the power
f^{\prime}\left(-1\right)=3\times 1
Multiply the numbers
f^{\prime}\left(-1\right)=3
\begin{align}&f^{\prime}\left(-1\right)=3\\&x_{2}=1\end{align}
Find the values of the derivatives for the chosen point
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Evaluate
f^{\prime}\left(1\right)=3\times 1^{2}
Evaluate the power
f^{\prime}\left(1\right)=3\times 1
Multiply the numbers
f^{\prime}\left(1\right)=3
\begin{align}&f^{\prime}\left(-1\right)=3\\&f^{\prime}\left(1\right)=3\end{align}
\begin{align}&x\leq 0\text{ is increasing interval}\\&f^{\prime}\left(1\right)=3\end{align}
\begin{align}&x\leq 0\text{ is increasing interval}\\&x\geq 0\text{ is increasing interval}\end{align}
Solution
\begin{align}&\text{The increasing interval is }x \in \mathbb{R}\\&\textrm{No decreasing interval}\end{align}
Find the range
f\left(x\right) \in \mathbb{R}
Evaluate
f\left(x\right)=x^{3}
Solution
f\left(x\right) \in \mathbb{R}
Find the vertical asymptotes
\text{No vertical asymptotes}
Evaluate
f\left(x\right)=x^{3}
Find the domain
f\left(x\right)=x^{3},x \in \mathbb{R}
Solution
\text{No vertical asymptotes}
Find the horizontal asymptotes
\text{No horizontal asymptotes}
Evaluate
f\left(x\right)=x^{3}
\text{To determine the horizontal asymptote,evaluate the limits }\lim _{x\rightarrow +\infty}\left(f\left(x\right)\right)\text{ and }\lim _{x\rightarrow -\infty}\left(f\left(x\right)\right)
\begin{align}&\lim _{x\rightarrow +\infty}\left(x^{3}\right)\\&\lim _{x\rightarrow -\infty}\left(x^{3}\right)\end{align}
Calculate
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Evaluate
\lim _{x\rightarrow +\infty}\left(x^{3}\right)
Rewrite the expression
\left(\lim _{x\rightarrow +\infty}\left(x\right)\right)^{3}
Calculate
\left(+\infty\right)^{3}
Calculate
+\infty
\begin{align}&+\infty\\&\lim _{x\rightarrow -\infty}\left(x^{3}\right)\end{align}
Calculate
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Evaluate
\lim _{x\rightarrow -\infty}\left(x^{3}\right)
Rewrite the expression
\left(\lim _{x\rightarrow -\infty}\left(x\right)\right)^{3}
Calculate
\left(-\infty\right)^{3}
Calculate
-\infty
\begin{align}&+\infty\\&-\infty\end{align}
Solution
\text{No horizontal asymptotes}
Find the oblique asymptotes
\text{No oblique asymptotes}
Evaluate
f\left(x\right)=x^{3}
\text{To determine the slope of the oblique asymptote,evaluate the limit }\lim _{x\rightarrow +\infty}\left(\frac{f\left(x\right)}{x}\right)\text{ and }\lim _{x\rightarrow -\infty}\left(\frac{f\left(x\right)}{x}\right)
\begin{align}&\lim _{x\rightarrow +\infty}\left(\frac{x^{3}}{x}\right)\\&\lim _{x\rightarrow -\infty}\left(\frac{x^{3}}{x}\right)\end{align}
Calculate
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Evaluate
\lim _{x\rightarrow +\infty}\left(\frac{x^{3}}{x}\right)
Rewrite the expression
\lim _{x\rightarrow +\infty}\left(\frac{x^{3}}{x}\times \frac{1}{1}\right)
Calculate
\lim _{x\rightarrow +\infty}\left(x^{2}\times \frac{1}{1}\right)
Calculate
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Calculate
\lim _{x\rightarrow +\infty}\left(x^{2}\right)\times \frac{\lim _{x\rightarrow +\infty}\left(1\right)}{\lim _{x\rightarrow +\infty}\left(1\right)}
Rewrite the expression
\lim _{x\rightarrow +\infty}\left(x^{2}\right)\times \frac{1}{\lim _{x\rightarrow +\infty}\left(1\right)}
Rewrite the expression
\lim _{x\rightarrow +\infty}\left(x^{2}\right)\times \frac{1}{1}
Rewrite the expression
\left(+\infty\right)\times \frac{1}{1}
Calculate
\left(+\infty\right)\times 1
Calculate
+\infty
+\infty
\begin{align}&+\infty\\&\lim _{x\rightarrow -\infty}\left(\frac{x^{3}}{x}\right)\end{align}
Calculate
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Evaluate
\lim _{x\rightarrow -\infty}\left(\frac{x^{3}}{x}\right)
Rewrite the expression
\lim _{x\rightarrow -\infty}\left(\frac{x^{3}}{x}\times \frac{1}{1}\right)
Calculate
\lim _{x\rightarrow -\infty}\left(x^{2}\times \frac{1}{1}\right)
Calculate
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Calculate
\lim _{x\rightarrow -\infty}\left(x^{2}\right)\times \frac{\lim _{x\rightarrow -\infty}\left(1\right)}{\lim _{x\rightarrow -\infty}\left(1\right)}
Rewrite the expression
\lim _{x\rightarrow -\infty}\left(x^{2}\right)\times \frac{1}{\lim _{x\rightarrow -\infty}\left(1\right)}
Rewrite the expression
\lim _{x\rightarrow -\infty}\left(x^{2}\right)\times \frac{1}{1}
Rewrite the expression
\left(+\infty\right)\times \frac{1}{1}
Calculate
\left(+\infty\right)\times 1
Calculate
+\infty
+\infty
\begin{align}&+\infty\\&+\infty\end{align}
Since the slope of the oblique asymptote is not defined,the function has no oblique asymptote
\begin{align}&\text{No oblique asymptotes}\\&+\infty\end{align}
Since the slope of the oblique asymptote is not defined,the function has no oblique asymptote
\begin{align}&\text{No oblique asymptotes}\\&\text{No oblique asymptotes}\end{align}
Solution
\text{No oblique asymptotes}
Determine if even, odd or neither
\text{Odd}
Evaluate
f\left(x\right)=x^{3}
\text{Substitute }-x\text{ for }x
f\left(-x\right)=\left(-x\right)^{3}
Simplify
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Evaluate
\left(-x\right)^{3}
To raise a product to a power,raise each factor to that power
\left(-1\right)^{3}\times x^{3}
Evaluate the power
-x^{3}
f\left(-x\right)=-x^{3}
\text{Substitute }f\left(x\right)\text{ for }x^{3}
f\left(-x\right)=-f\left(x\right)
Solution
\text{Odd}
Find the stationary points
\left(0,0\right)
Evaluate
f\left(x\right)=x^{3}
Solution
\left(0,0\right)
Find the inflection points
\left(0,0\right)
Evaluate
f\left(x\right)=x^{3}
Solution
\left(0,0\right)
Choose Method
Find the inverse
Evaluate the derivative
Find the domain
\text{Find the }x\text{-intercept/zero}
Find the y-intercept
Find the critical numbers
Find the local extrema
Find the increasing or decreasing interval
Find the range
Find the vertical asymptotes
Find the horizontal asymptotes
Find the oblique asymptotes
Determine if even, odd or neither
Find the stationary points
Find the inflection points
Graph